On generators and relations for $O_n^\circ(V)$ in higher-level Zhu algebras, Part II

Abstract

We study circle relations in the higher-level Zhu algebra of a vertex operator algebra $V$. For regular vacuum words in homogeneous letters, we prove the termwise correction-residue conjecture of Part I: every correction in the one-sided iterate expansion belongs separately to the finite-length plus subspace. The first correction formed with the left truncated field vanishes identically. Multiplying the diagonal commutator kernel by a power of $1+x$ turns it into a Laurent polynomial. This proves the vanishing and writes every other correction as a finite combination of product residues with fewer $Y_L$ factors, coming from the word obtained by replacing one negative mode by a nonnegative one. These results hold at all levels $n\ge0$, without an auxiliary-degree hypothesis. For $n\ge1$ and a homogeneous ordered pair $(u,v)$, two explicit triangular families span the quadratic residue operators modulo the length-one and singular-iterate relations, giving $\dim\mathcal{Q}_n((u,v))\le n(n+1)$. The universal product-coefficient arrays have dimension exactly $n(n+1)$ and three natural bases; restriction to the regular mode square is an isomorphism. This proves operator spanning and gives invertible matrices for two complementary diagonal eliminations. For equal inputs, mode interchange reduces the families to their odd-indexed members, giving $\dim\mathcal{Q}_n((u,u))\le 2\lfloor (n+1)^2/4\rfloor$. The pairwise estimates give an explicit global quadratic bound for any finite homogeneous strong generating set. Heisenberg, Virasoro, and affine $\mathfrak{sl}_2$ examples illustrate equality and strict inequality in the operator bounds.